Is it okay to use "month-old" without "a" or "one" in front of it?
Yes, it is okay to use "month-old" without an article in front of it because the hyphen turns it into an adjective in this case. If you were to use "month old" without a hyphen, "month" would still be a noun, and so it would require there to be an article in front of it. But a hyphen links the two separate words of "month" and "old" together to form an adjective, so there is no need for an article to come in front of it. Ex: I bought this month-old car at the dealership yesterday. Ex: Her month-old child is tiny.